A short proof of Helson's conjecture

Let $\alpha \text{\hspace{0.17em}⁣}:\mathbb{N}\to S^1$ be the Steinhaus multiplicative function: a completely multiplicative function such that $(\alpha (p){)}_{p\text{ prime}}$ are i.i.d.~random variables uniformly distributed on the complex unit circle $S^1$. Helson conjectured that $\mathbb{E}\mathrm{\mid }{\sum }_{n\le x}\alpha (n)\mathrm{\mid }=o(\sqrt{x})$ as $x\to \mathrm{\infty }$, and this was solved in a strong form by Harper. We give a short proof of the conjecture using a result of Saksman and Webb on a random model for the zeta function.